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<b>let </b><font color="Maroon">C</font> be   <a href="cat_4.html#NM1">Cartesian_category</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>





<b>set </b><font color="Maroon">p1</font> = the <a href="cat_4.html#U3" target="_self">Proj1</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">p2</font> = the <a href="cat_4.html#U4" target="_self">Proj2</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E1:28"/>
the <a href="cat_4.html#U2" target="_self">CatProd</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="cat_3.html#R2">is_a_product_wrt</a> the <a href="cat_4.html#U3" target="_self">Proj1</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="cat_4.html#U4" target="_self">Proj2</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:28"/><b>then </b>
<font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">b</font> <a href="cat_3.html#R2">is_a_product_wrt</a> the <a href="cat_4.html#U3" target="_self">Proj1</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="cat_4.html#U4" target="_self">Proj2</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 
;<br/>
<a NAME="E3:28"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default">the <a href="cat_4.html#U4" target="_self">Proj2</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default">the <a href="cat_4.html#U4" target="_self">Proj2</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="cat_3.html#D16">CAT_3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K4" target="_self">pr2</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_4.html#K4" target="_self">pr2</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>


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